https://file.aiquickdraw.com/mathbot/file/1726434195620k77suebn.jpg

The equation in the image is:

\[
2^x \cdot 3^x = 216
\]

To solve this for \( x \):

### Step 1: Rewrite the equation
We can factor the left-hand side:
\[
(2 \cdot 3)^x = 216
\]
Simplifying inside the parentheses:
\[
6^x = 216
\]

### Step 2: Express 216 as a power of 6
We need to check if 216 can be written as \( 6^n \). Since:
\[
6^3 = 216
\]
This means the equation becomes:
\[
6^x = 6^3
\]

### Step 3: Solve for \( x \)
Since the bases are the same, we can equate the exponents:
\[
x = 3
\]

Thus, the solution is:
\[
x = 3
\]

Would you like any further details or explanations?

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Here are 5 related questions:
1. How would you solve \( 2^x \cdot 5^x = 500 \)?
2. What is the general method to solve exponential equations of the form \( a^x \cdot b^x = c \)?
3. How would you approach the equation \( 2^x + 3^x = 216 \)?
4. Can you solve \( 6^x = 432 \) for \( x \)?
5. What are logarithms, and how are they useful for solving exponential equations?

**Tip:** When solving exponential equations, try to express both sides with the same base for easier comparison.

Learn how to solve the exponential equation 2^x . 3^x = 216 by factoring and using the law of exponents. Step-by-step solution provided for high school students in Grades 9-11.

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